Ambassador to the humans

Different definitions get used in different contexts. I prefer the measure theoretical definition which mostly matches up with that first definition you give. It's basically just a function that allows us to map an element of some set (be it humans, pigs, voters, etc...) to some element of the real number line. We typically don't require any sort of probability measure to be assigned directly to the random variables. Probability can be derived from these once we assign a probability measure to the original set. This might seem overly complicated but it actually is quite freeing and allows us to think much more naturally about how we actually deal with these types of problems in the real world.